Internal problem ID [12902]
Internal file name [OUTPUT/11555_Tuesday_November_07_2023_11_27_00_PM_5441983/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number: 4.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }-4 y^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{4 y^{2}}d y &= t +c_{1}\\ -\frac {1}{4 y}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {1}{4 \left (t +c_{1} \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{4 \left (t +c_{1} \right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{4 \left (t +c_{1} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-4 y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=4 y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=4 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d t =\int 4d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=4 t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{4 t +c_{1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 11
dsolve(diff(y(t),t)=4*y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {1}{-4 t +c_{1}} \]
✓ Solution by Mathematica
Time used: 0.157 (sec). Leaf size: 20
DSolve[y'[t]==4*y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {1}{4 t+c_1} \\ y(t)\to 0 \\ \end{align*}